Because you could, in theory, lie about thing, does not mean that thing is badly defined. You could lie about your ability to speak German, whether you are a mother or not, what your sexuality or religious belief is. That does not reflect whether those concepts are valid. That is not to say that people lying about their gender identity could not be dangerous.
Of course you can lie about anything. But the difference with your construct of who is a woman or a man, and the examples you give above about lying, is that for these others you have some form of actual definition against which to verify a statement. If I said I can speak German, you can test my claim by speaking German to me. Similarly, if I am claiming to have a religious belief, you can ask me about the religion or check membership with a church. If I am claiming something about my sexuality, you can judge it by my history of sexual partners, etc.
This is absolutely not the case in what you're saying about what it means to be a woman. If your definition of "woman" is just whoever says "yes" to the question "are you a woman?", how are you going to test that? It literally means absolutely whoever says "yes." So, in fact, it becomes impossible to lie about womanhood with your definition. The mere act of saying "yes," regardless of feeling and motivation, IS womanhood to you because it will always meet your definition satisfactorily. ANYONE who says "yes" must be a woman. How can you say otherwise? This will include those individuals who may be saying "yes" with genuinely ill intent.
Can you really not see the problem with this? Again, how do I know if I am a woman in order to answer your question with a "yes" or "no"?
What is a woman?